I have two matries given by (M' means derivative w.r.t x)
$ M=\left( \begin{array}{ccc} f_1(x) & f_2(x) & f_3(x) \\ f_4(x) & f_5(x)& f_6(x) \\ f_7(x) & f_8(x) & f_9(x) \\ \end{array} \right) \tag 1$
$ M'=\left( \begin{array}{ccc} f_1'(x) & f_2'(x) & f_3'(x) \\ f_4'(x) & f_5'(x)& f_6'(x) \\ f_7'(x) & f_8'(x) & f_9'(x) \\ \end{array} \right) \tag 2$ . These $f_i(x)$ can be any type of functions. They can be equal ($f_i=f_j$)or different or equal some of them and different others ,they can be constant functions. There is no limit or condition at all. Only one exception is that total matrix M cant be a zero matrix with full entries as zeros.
Question
Can you provide all $f_i$( any function you can select) such that it just satisfies following condition(all $a_0,b_0,c_0,a_1,b_1,c_1$ are constants cant be altered,but x is a variable) . or is it not possible to have such one M? please clarify
$ M'=\{\left( \begin{array}{ccc} 0 & -c_0 & b_0 \\ c_0 & 0 & -a_0 \\ -b_0 & a_0 & 0 \\ \end{array} \right)+\left( \begin{array}{ccc} 0 & -(c_1-c_0) & (b_1-b_0) \\ (c_1-c_0) & 0 & -(a_1-a_0) \\ -(b_1-b_0) & (a_1-a_0) & 0 \\ \end{array} \right)x\}M \tag 3$
a,b,c are constants. You can select any arbitrar function. I tried a lot. Since there is no standard methods. I couldnt get a proper one.Thanks a lot
Not a solution but maybe a step in the right direction. First you can formulate your problem as a matrix differential equation. With the matrix vectorization operator, which stacks the columns of the matrix , we write:
$$m(x) = \text{vec}(M(x))$$
whereas $m(x)$ is just the vector of your unknown functions $f_i(x)$. Your original problem is
$$M'(x) = (A + Bx)M(x)$$
which transforms to the matrix differential equation, after applying matrix vectorization on both sides
$$m'(x) = ((I \otimes A) + (I \otimes Bx))m(x)$$
with the Kronecker matrix product $\otimes$. So we have an matrix differential equation of the form
$$m'(x) = (\widetilde A + \widetilde Bx)m(x)$$
with some constant matrices $\widetilde A$ and $\widetilde B$ that essentially just contain the numbers $a_0, b_0$ etc.
This is solvable, when the matrices $\widetilde A$ and $\widetilde B$ are commuting, i.e. $\widetilde A \widetilde B = \widetilde B \widetilde A$ using the matrix exponential:
$$m(x) = \exp (\widetilde A x + 1/2\widetilde Bx^2)$$
However, I dont know whether the matrices commute (which is simple to check, but tedious) and unfortunately I have no solution when the do not commute.
This Math24 page gives some background information, on how to solve matrix differential equations with variable coefficients (as in this case).